The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance

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Abstract
The linear stability of two superimposed layers of fluid, heated from below and separated by a thin conducting plate, is investigated. It is shown that when the ratio of the depths of the layers is close to 1:2, two distinct modes of convection can occur with preferred horizontal wavenumber in the ratio 1:2. The nonlinear evolution of a disturbance consisting of both modes is considered, and it is shown that travelling waves are the preferred mode of nonlinear convection for a wide range of parameter values. Other possible types of behaviour, including modulated waves and an attracting homoclinic trajectory, are also described in detail.